We perform a projection study for the sensitivity of the upgraded CMS detector to new physics in a mono-Higgs sector, h $\to$ bb plus missing transverse momentum final state, at the HL-LHC. The focus lies on signals with boosted Higgs bosons, resulting in a large radius jet containing both b quarks that can are identified as originating from a Higgs boson. The search is sensitive to a variety of BSM models and in this analysis the results are interpreted in terms of 2HDM+a model. We find that with a data set corresponding to 3000 fb$^{-1}$, $m_{\text{A}}$ masses from 750 to 2000 GeV could be excluded for $m_{\text{a}} = 250$ GeV and $m_{\text{A}}$ masses from 1250 to 1600 GeV could be excluded for $m_{\text{a}} = 500$ GeV. The 2HDM+a signals with $m_{\text{a}} = 250$ and $m_{\text{A}}= 1000-1600$ GeV, which are not yet excluded, could reach a significance near 5$\sigma$.

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